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Theorem isnumbasgrplem2 41417
Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
isnumbasgrplem2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 17087 . . 3 Base Fn V
2 ssv 3968 . . 3 Grp ⊆ V
3 fvelimab 6914 . . 3 ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
41, 2, 3mp2an 690 . 2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5 harcl 9495 . . . . . 6 (har‘𝑆) ∈ On
6 onenon 9885 . . . . . 6 ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
75, 6ax-mp 5 . . . . 5 (har‘𝑆) ∈ dom card
8 xpnum 9887 . . . . 5 (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
97, 7, 8mp2an 690 . . . 4 ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10 ssun1 4132 . . . . . . . 8 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11 simpr 485 . . . . . . . 8 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
1210, 11sseqtrrid 3997 . . . . . . 7 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13 fvex 6855 . . . . . . . 8 (Base‘𝑥) ∈ V
1413ssex 5278 . . . . . . 7 (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
1512, 14syl 17 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
167a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17 simp1l 1197 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18123ad2ant1 1133 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19 simp2 1137 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎𝑆)
2018, 19sseldd 3945 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21 ssun2 4133 . . . . . . . . . . 11 (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
2221, 11sseqtrrid 3997 . . . . . . . . . 10 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23223ad2ant1 1133 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24 simp3 1138 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
2523, 24sseldd 3945 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26 eqid 2736 . . . . . . . . 9 (Base‘𝑥) = (Base‘𝑥)
27 eqid 2736 . . . . . . . . 9 (+g𝑥) = (+g𝑥)
2826, 27grpcl 18756 . . . . . . . 8 ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
2917, 20, 25, 28syl3anc 1371 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
30 simp1r 1198 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
3129, 30eleqtrd 2840 . . . . . 6 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32 simplll 773 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
3322ad2antrr 724 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34 simprl 769 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
3533, 34sseldd 3945 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36 simprr 771 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
3733, 36sseldd 3945 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
3812ad2antrr 724 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39 simplr 767 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎𝑆)
4038, 39sseldd 3945 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
4126, 27grplcan 18809 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
4232, 35, 37, 40, 41syl13anc 1372 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
43 simplll 773 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑥 ∈ Grp)
4412ad2antrr 724 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45 simprr 771 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑𝑆)
4644, 45sseldd 3945 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑 ∈ (Base‘𝑥))
47 simprl 769 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎𝑆)
4844, 47sseldd 3945 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎 ∈ (Base‘𝑥))
4922ad2antrr 724 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50 simplr 767 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (har‘𝑆))
5149, 50sseldd 3945 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (Base‘𝑥))
5226, 27grprcan 18784 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
5343, 46, 48, 51, 52syl13anc 1372 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
54 harndom 9498 . . . . . . 7 ¬ (har‘𝑆) ≼ 𝑆
5554a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
5615, 16, 16, 31, 42, 53, 55unxpwdom3 41408 . . . . 5 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆* ((har‘𝑆) × (har‘𝑆)))
57 wdomnumr 10000 . . . . . 6 (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
589, 57ax-mp 5 . . . . 5 (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
5956, 58sylib 217 . . . 4 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60 numdom 9974 . . . 4 ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
619, 59, 60sylancr 587 . . 3 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
6261rexlimiva 3144 . 2 (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
634, 62sylbi 216 1 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  cun 3908  wss 3910   class class class wbr 5105   × cxp 5631  dom cdm 5633  cima 5636  Oncon0 6317   Fn wfn 6491  cfv 6496  (class class class)co 7357  cdom 8881  harchar 9492  * cwdom 9500  cardccrd 9871  Basecbs 17083  +gcplusg 17133  Grpcgrp 18748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-oi 9446  df-har 9493  df-wdom 9501  df-card 9875  df-acn 9878  df-nn 12154  df-slot 17054  df-ndx 17066  df-base 17084  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752
This theorem is referenced by:  isnumbasabl  41419  isnumbasgrp  41420
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